Abstract

The thick phase-diffraction grating (an elementary hologram) is considered which is obtained as a result of the interference of a plane or inhomogeneous object wave with an evanescent wave in the photographic emulsion resulting from the total reflection of a Gaussian reference beam at the emulsion surface. The linear theory of diffraction efficiencies in weak diffraction approximation for normally polarized light is formulated. Using a digital computer the geometry of the recording process is optimized, and the uncertainty factor of the conventional Bragg condition for volume diffraction gratings in the case of inhomogeneous waves is discussed.

© 1980 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  2. O. Bryngdahl, Prog. Opt. 11, 0000 (1973).
  3. O. Bryngdahl, J. Opt. Soc. Am. 59, 1645 (1969).
    [CrossRef]
  4. H. Nassenstein, Optik 29, 456 (1969).
  5. H. Nassenstein, Optik 29, 597 (1969).
  6. H. Nassenstein, Optik 30, 44, 201 (1969).
  7. H. Nassenstein, Opt. Commun. 1, 146 (1969); H. Nassenstein, Opt. Commun. 2, 231(1970).
    [CrossRef]
  8. H. Nassenstein, Natürwissenschaften 57, 468 (1970).
  9. W. Lukosz, A. Wüthrich, Optik 41, 191 (1974).
  10. A. Wüthrich, W. Lukosz, Optik 42, 315 (1975).
  11. W. Lukosz, A. Wüthrich, Opt. Commun. 19, 232 (1976).
    [CrossRef]
  12. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  13. J. Woźnicki, “Evanescent Wave Structure for Total Reflection of Gaussian Beam at a Plane Interface,” Opt. Appl., 00, 0000 (19xx) in print.
  14. J. Woźnicki, “Reference Gaussian Beam in Evanescent Wave Holography,” Optik00, 0000 (19xx), in print.
  15. See, for example, P. Roman, Advanced Quantum Theory (Addison Wesley, Reading, Mass., 1965), p. 155.
  16. J. Woźnicki, Ph.D. Dissertation, Warsaw Technical U. (1978).
  17. As an example, n2 = 1.63 − i × 10−3 for Agfa-Gevaert Scientia 8E75 emulsion can be considered.
  18. W. Braunbek, Z. Naturforsch. Teil A: 6a, 12 (1951).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  20. A. A. Michelson, H. G. Gale, Nature London 115, 566 (1925).
    [CrossRef]
  21. D. Gabor, Proc. R. Soc. London Ser. A: 197, 454 (1949); Proc. Phys. Soc. London Sect. B 64, 449 (1951).
    [CrossRef]

1976

W. Lukosz, A. Wüthrich, Opt. Commun. 19, 232 (1976).
[CrossRef]

1975

A. Wüthrich, W. Lukosz, Optik 42, 315 (1975).

1974

W. Lukosz, A. Wüthrich, Optik 41, 191 (1974).

1973

O. Bryngdahl, Prog. Opt. 11, 0000 (1973).

1970

H. Nassenstein, Natürwissenschaften 57, 468 (1970).

1969

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

O. Bryngdahl, J. Opt. Soc. Am. 59, 1645 (1969).
[CrossRef]

H. Nassenstein, Optik 29, 456 (1969).

H. Nassenstein, Optik 29, 597 (1969).

H. Nassenstein, Optik 30, 44, 201 (1969).

H. Nassenstein, Opt. Commun. 1, 146 (1969); H. Nassenstein, Opt. Commun. 2, 231(1970).
[CrossRef]

1966

1951

W. Braunbek, Z. Naturforsch. Teil A: 6a, 12 (1951).

1949

D. Gabor, Proc. R. Soc. London Ser. A: 197, 454 (1949); Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[CrossRef]

1925

A. A. Michelson, H. G. Gale, Nature London 115, 566 (1925).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Braunbek, W.

W. Braunbek, Z. Naturforsch. Teil A: 6a, 12 (1951).

Bryngdahl, O.

O. Bryngdahl, Prog. Opt. 11, 0000 (1973).

O. Bryngdahl, J. Opt. Soc. Am. 59, 1645 (1969).
[CrossRef]

Gabor, D.

D. Gabor, Proc. R. Soc. London Ser. A: 197, 454 (1949); Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[CrossRef]

Gale, H. G.

A. A. Michelson, H. G. Gale, Nature London 115, 566 (1925).
[CrossRef]

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

Li, T.

Lukosz, W.

W. Lukosz, A. Wüthrich, Opt. Commun. 19, 232 (1976).
[CrossRef]

A. Wüthrich, W. Lukosz, Optik 42, 315 (1975).

W. Lukosz, A. Wüthrich, Optik 41, 191 (1974).

Michelson, A. A.

A. A. Michelson, H. G. Gale, Nature London 115, 566 (1925).
[CrossRef]

Nassenstein, H.

H. Nassenstein, Natürwissenschaften 57, 468 (1970).

H. Nassenstein, Optik 29, 456 (1969).

H. Nassenstein, Optik 29, 597 (1969).

H. Nassenstein, Optik 30, 44, 201 (1969).

H. Nassenstein, Opt. Commun. 1, 146 (1969); H. Nassenstein, Opt. Commun. 2, 231(1970).
[CrossRef]

Roman, P.

See, for example, P. Roman, Advanced Quantum Theory (Addison Wesley, Reading, Mass., 1965), p. 155.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Woznicki, J.

J. Woźnicki, Ph.D. Dissertation, Warsaw Technical U. (1978).

J. Woźnicki, “Evanescent Wave Structure for Total Reflection of Gaussian Beam at a Plane Interface,” Opt. Appl., 00, 0000 (19xx) in print.

J. Woźnicki, “Reference Gaussian Beam in Evanescent Wave Holography,” Optik00, 0000 (19xx), in print.

Wüthrich, A.

W. Lukosz, A. Wüthrich, Opt. Commun. 19, 232 (1976).
[CrossRef]

A. Wüthrich, W. Lukosz, Optik 42, 315 (1975).

W. Lukosz, A. Wüthrich, Optik 41, 191 (1974).

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

J. Opt. Soc. Am.

Nature London

A. A. Michelson, H. G. Gale, Nature London 115, 566 (1925).
[CrossRef]

Natürwissenschaften

H. Nassenstein, Natürwissenschaften 57, 468 (1970).

Opt. Commun.

H. Nassenstein, Opt. Commun. 1, 146 (1969); H. Nassenstein, Opt. Commun. 2, 231(1970).
[CrossRef]

W. Lukosz, A. Wüthrich, Opt. Commun. 19, 232 (1976).
[CrossRef]

Optik

W. Lukosz, A. Wüthrich, Optik 41, 191 (1974).

A. Wüthrich, W. Lukosz, Optik 42, 315 (1975).

H. Nassenstein, Optik 29, 456 (1969).

H. Nassenstein, Optik 29, 597 (1969).

H. Nassenstein, Optik 30, 44, 201 (1969).

Proc. R. Soc. London Ser. A:

D. Gabor, Proc. R. Soc. London Ser. A: 197, 454 (1949); Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[CrossRef]

Prog. Opt.

O. Bryngdahl, Prog. Opt. 11, 0000 (1973).

Z. Naturforsch. Teil A

W. Braunbek, Z. Naturforsch. Teil A: 6a, 12 (1951).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

J. Woźnicki, “Evanescent Wave Structure for Total Reflection of Gaussian Beam at a Plane Interface,” Opt. Appl., 00, 0000 (19xx) in print.

J. Woźnicki, “Reference Gaussian Beam in Evanescent Wave Holography,” Optik00, 0000 (19xx), in print.

See, for example, P. Roman, Advanced Quantum Theory (Addison Wesley, Reading, Mass., 1965), p. 155.

J. Woźnicki, Ph.D. Dissertation, Warsaw Technical U. (1978).

As an example, n2 = 1.63 − i × 10−3 for Agfa-Gevaert Scientia 8E75 emulsion can be considered.

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Figures (5)

Fig. 1
Fig. 1

Reference Gaussian beam in the local coordinate system (ζ,y,ξ).

Fig. 2
Fig. 2

Geometry of recording and reconstruction of a hologram. Symbols with primes describe the field in the surrounding medium, symbols without primes refer to photographic emulsions. Critical angles φ0 and φ0c are shown by dashed lines. U p represents the plane object wave, U e the Gaussian reference beam, U c the plane reconstructing wave, and U d the reconstructed beam (which appears in the text with appropriate indices). Subscripts 1 and 2 denote the wave connected with an appropriate interference term δɛ1,2 of the modulated hologram structure; (+) and (−) signs specify the propagation direction of the reconstructed wave z > z0 or z < 0, respectively.

Fig. 3
Fig. 3

Diffraction efficiencies η of the holograms, recorded and reconstructed in the configuration shown at the top of the figure, vs the angle of incidence φp1 of object wave U p . The same type of line has been used for both the reconstructed wave and the curve to which it refers. Emulsion shrinkage has been taken into account; in all cases φ1 = 71° > φ0 was used. The angle of the incidence of the reconstructing wave was chosen to be optimal: φc1 = φ0c or φc1 = −φ0c, respectively.

Fig. 4
Fig. 4

Variations of diffraction efficiency η of holograms recorded and reconstructed in the configuration shown at the top of the figure vs the angle of incidence φ1 of the Gaussian reference beam for: (a) evanescent φp1 = 75°, (b) plane φp1 = 10°; object wave Up in optimum reconstruction conditions. The same type of line has been used for both the reconstructed wave and the curve to which it refers. Emulsion shrinkage has been taken into account.

Fig. 5
Fig. 5

Diffraction efficiencies η of the holograms recorded and reconstructed in the configuration shown at the top of the figure vs the difference in wavelength in the reconstruction and recording process. The same type of line has been used for both the reconstructed wave and the curve to which it refers. The figure shows pairs of curves with (s = 1.1) and without (s = 1.0) the emulsion shrinkage effect. In all cases φ1 = 71° > φ0 was used. The angle of incidence of the reconstructing wave was chosen to be optimal: φc1 = φ0c or φ c 1 = φ 0 c Δ λ = 0 as shown in the figure.

Equations (20)

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U e ( x , y , z ) = U e w 0 w ( ξ ) exp [ - ( α ζ ) 2 + ( β y ) 2 [ w ( ξ ) ] 2 ] × exp { - i [ k e ξ - Φ ( ξ ) + ( α ζ ) 2 + ( β y ) 2 2 R ( ξ ) k e ] } ,
U e ( x , y , z ) = C - + V e ( x , y , z ; p , q ) d p d q C - + V e ( p , q ) exp { i [ k e , x ( p , q ) x + k e , y ( p , q ) y + k e , z ( p , q ) z ] } dpdq ,
C = ( k e 2 π ) 2 π α β b U e w 0 w ( ξ 1 ) exp { - i [ k e ξ 1 - Φ ( ξ 1 ) ] } , b = 1 [ w ( ξ 1 ) ] 2 + i k e 2 R ( ξ 1 ) ,
V e ( p , q ) = T ( p , q ) exp { - k e 4 b [ ( p α ) 2 + ( q β ) 2 ] } , T ( p , q ) = 2 ( p sin φ 1 + m cos φ 1 ) [ 1 - n 2 2 1 + n 2 2 1 ( p sin φ 1 + m cos φ 1 ) 2 ] 1 / 2 + p sin φ 1 + m cos φ 1
k e , x ( p , q ) = k e ( p cos φ 1 - m sin φ 1 ) , k e , y ( p , q ) = k e q , k e , z ( p , q ) = k e 1 n 21 [ 1 - n 2 2 1 + n 2 2 1 ( p sin φ 1 + m cos φ 1 ) 2 ] 1 / 2
I = 1 2 [ U p ( x , z ) + V e ( x , y , z ; p , q ) ] * [ U p ( x , z ) + V e ( x , y , z ; p , q ) ]
δ ɛ 1 ( x , y , z ; p , q ) = 1 2 d ɛ d I V e ( p , q ) U p * exp ( i { [ k e , x ( p , q ) - k p , x ] x + k e , y ( p , q ) y + [ k e , z ( p , q ) - k p , z * ] z } ) ,
δ ɛ 2 ( x , y , z ; p , q ) = 1 2 d ɛ d I V e * ( p , q ) U p exp ( i { ( [ k p , x - k e , x ( p , q ) ] x - k e , y ( p , q ) y + [ k p , z - k e , z * ( p , q ) ] z ] }
[ Δ + ( ω c / c ) 2 ɛ ( x , y , z ; p , q ) ] U ( x , y , z ; p , q ) = 0.
V d 1 ( x , y , z = 0 z 0 ; p , q ) = ± ( ω c c ) 2 d ɛ d I U c U p * V e ( p , q ) · k c , z ( k c , z + k c , z ) ( k d , z + k d , z ) exp [ i ( k d , x x + k d , y y + k c , z z ) ] ( - k d , z + k e , z - k p , z * + k c , z ) ,
V d 2 ( x , y , z = 0 z 0 ; p , q ) = ± ( ω c c ) 2 d ɛ d I U c U p V e * ( p , q ) · k c , z ( k c , z + k c , z ) ( k d , z + k d , z ) exp [ i ( k d , x x + k d , y y + k d , z z ) ] ( - k d , z - k e , z * + k p , z + k c , z ) .
k d , x = k x ( p , q ) ;             k d , y = k y ( p , q ) ;             k d , z + = - k d , z - = k z ( p , q ) ,
k x 1 , 2 = k ± e , x ( p , q ) k p , x + k c , x , k y 1 , 2 = k ± e , y ( p , q ) , k z 1 , 2 = { ( k c 2 - k x 1 , 2 2 - k y 1 , 2 2 ) 1 / 2 when k x 1 , 2 2 + k y 1 , 2 2 k c 2 i ( k x 1 , 2 2 + k y 1 , 2 2 - k c 2 ) 1 / 2 when k x 1 , 2 2 + k y 1 , 2 2 > k c 2 .
U d 1 , 2 ( x , y , z = 0 z 0 ) = C - + V d 1 , 2 ( x , y , z = 0 z 0 ; p , q ) d q d q
Re { z [ F d 1 , 2 ( x , y , z = 0 z 0 ) ] } Re [ k d , z 1 , 2 ( p = 0 , q = 0 ) ] Re [ k d , z 1 , 2 ( 0 ) ] ,
P d 1 , 2 = F S z d 1 , 2 d x d y F S z ( 0 ) d 1 , 2 d x d y = 1 ω μ 0 Re [ k d , z 1 , 2 ( 0 ) ] · F U d 1 , 2 ( x , y , z z 0 0 ) 2 d x d y .
Δ ɛ ( 0 ) = ( d ɛ / d I ) V e ( p = 0 , q = 0 ) U p ( d ɛ / d I ) V e ( 0 ) U p
V e ( 0 ) ( p , q ) = V e ( p , q ) / V e ( 0 )
η 1 [ Δ ɛ ( 0 ) ] 2 = ( π α β b ) 2 ( ω c c ) 4 Re [ k d , z 1 ( 0 ) ] k c , z F k c , z + k c , z 2 F | - + V ρ ( 0 ) ( p , q ) · exp { i [ k d , x ( p , q ) x + k d , y ( p , q ) y + k d , z ( p , q ) z ] } d p d q [ k d , z ( p , q ) + k d , z ( p , q ) ] [ - k d , z ( p , q ) + k e , z ( p , q ) - k p , z * + k c , z ] | 2 d x d y ,
η 2 [ Δ ɛ ( 0 ) ] 2 = ( π α β b ) 2 ( ω c c ) 4 Re [ k d , z 2 ( 0 ) ] k c , z F k c , z + k c , z 2 F | - + V e ( 0 ) * ( p , q ) · exp { i [ k d , x ( p , q ) x + k d , y ( p , q ) y + k d , z ( p , q ) z ] } d p d q [ k d , z ( p , q ) + k d , z ( p , q ) ] [ - k d , z ( p , q ) + k e , z * ( p , q ) - k p , z + k c , z ] | 2 d x d y ,

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